Lecture: 3
Course: M339D/M389D - Intro to Financial Math
Page: 1 of 5
University of Texas at Austin
Lecture 3
Basic risk management. An introduction to forward contracts.
3.1. Basic risk management.
Definition 3.1. We say that a portfolio is long with respect to a certain asset if its payoff
function is increasing as a function of that asset’s final price. We say that a portfolio is short
with respect to a certain asset if its payoff function is decreasing as a function of that asset’s
final price.
Remark 3.2. Neither the increase nor the decrease above are in the strict sense.
Producers or manufacturers of goods and services (commodities) are going to be profitable
if the revenue from the sale of produced goods in the market exceeds the manufacturing costs.
This makes the producers have an inherent long position in their product. On the other
hand, the users or buyers of commodities benefit from decrease in the price of the good;
they have an inherent short position with respect to that commodity. It is oftentimes not
enough to optimize the manufacturing process and/or rely on classical insurance to keep a
company profitable. This is where financial hedging comes in.
3.2. Forward contracts. A forward contract is a binding agreement to buy/sell an underlying asset on an agreed-upon date in the future – the delivery date – at a fixed price set
today. The forward contract specifies the following:
• the asset to be delivered,
• the quantity of the asset to be delivered,
• the delivery logistics: the time, date, place, e.g.,
• the price the buyer will pay at the time of delivery: the forward price,
• type of settlement: cash settlement or physical delivery (the former is preferable
due to simplicity and lower transaction costs).
The person whose position is the long forward will be required to “buy” the underlying
asset for the forward price. So, his/her payoff function is
v(s) = s − F,
where F denotes the forward price. So, with S(T ) denoting the final price of the underlying
asset, the realized payoff for a long forward contract equals
S(T ) − F.
Analogously, if one has the short forward, one’s payoff is
F − S(T ).
The payoff graphs for both positions are displayed below. The blue line is the long forward
payoff, and the red line is the short forward payoff. The forward price is set to be F = 1000.
Instructor: Milica Čudina
Lecture: 3
Course: M339D/M389D - Intro to Financial Math
Page: 2 of 5
1000
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3.2.1. Forward contract versus outright purchase. The natural question to ask at this point
is:
“What is the difference between the outright purchase of the underlying asset and the long
forward contract on the same asset?
First, let us go back to the way that the forward contract is agreed upon by the two parties.
At no point in the discussion above did we mention the initial cost of the forward contract.
The reason is simple – there is no initial cost. For a forward contract, the only action that
happens at time−0 is the agreement reached which includes the stipulations in the forward
contract. However, there are no cash-flows taking place at time−0. In particular, this means
that the profit and the payoff curves coincide in the case of the forward contract.
Second, let us focus on the forward price F briefly. Imagine that the party with the log
forward contract wants to be certain that he/she has sufficient funds at the delivery date T
to pay the forward price. One way of ensuring this would be an investment at time−0 in
a zero-coupon bond with maturity date T and redemption amount F . We can just think
about this step as if he/she simply made a deposit into a savings account today and did
nothing until the delivery date – then he/she withdraws the balance and uses it to purchase
the underlying asset. With the purchase of the bond coupled with the long forward contract,
one creates a simple portfolio. Let’s try to see what the payoff curve for this portfolio is.
Again, let F = 1000 for the sake of being able to have a concrete graph.
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1000
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-1000
The blue line is the payoff of the long forward contract, and the red line is the payoff of
the bond. The payoff of the entire portfolio is the sum of the payoffs of its components –
the green line. If the green line reminds you of something, this is not accidental. In fact,
Instructor: Milica Čudina
Lecture: 3
Course: M339D/M389D - Intro to Financial Math
Page: 3 of 5
we encountered the same payoff curve in the previous lecture. It corresponded to the long
stock position – the simple outright purchase of a share of stock.
We have to emphasize that looking at the profit curve is not entirely straightforward. We
have to take into account that the owner of the stock is entitled to any dividend payments.
On the other hand, the holder of the forward contract does not receive dividend payments.
We will return to these issues soon when we discuss the ways to come up with a fair forward
price.
3.2.2. Credit risk. The term credit risk signifies the risk associated with the possibility that
one of the counterparties in the contract (any contract – not just a forward contract) might
fail to fulfill its financial obligations. To alleviate this risk and provide a sense of security needed to conduct the trades in the first place several steps might be taken. One
example is “standardization”– certain types of contracts are popular/useful enough to
become exchange-traded. There are two major consequences of this step: the contracts get
very detailed and standard (collateral requirements are introduced, e.g.), and the exchange
guarantees the transactions in conducts. However, not all contracts are traded frequently
enough to warrant the above approach. The blanket term for the contracts that are not
exchange traded is over-the-counter (OTC) contracts. In their context, remedies for credit
risk may include:
• credit checks,
• collateral,
• bank letter of credit, . . .
Problem 3.1. Source: Sample FM(DM) Problem #10.
Suppose stock XYZ has a current price of 100. The forward price for delivery of this stock
in 1 year is 110. Also, assume there are no dividends, and that the annual effective interest
rate is 10%, unless otherwise indicated (in the sequel).
Which of the following statements is FALSE?
A. The time−1 profit diagram and the time−1 payoff diagram for long positions in this
forward contract are identical.
B. The time−1 profit for a long position in this forward contract is exactly opposite to
the time−1 profit for the corresponding short forward position.
C. There is no comparative advantage to investing in the stock versus investing in the
forward contract.
D. If the 10% interest rate were continuously compounded instead of annual effective,
then it would be more beneficial to invest in the stock, rather than the forward
contract.
E. If there was a dividend of 3.00 paid 6 months from now, then it would be more
beneficial to invest in the stock, rather than the forward contract.
Solution:
A. TRUE
There is no initial cost to entering a forward contract, so its payoff and profit are
always equal.
B. TRUE
This statement is always true for long and short positions in the same contract.
C. TRUE
The profits are equal.
Instructor: Milica Čudina
Lecture: 3
Course: M339D/M389D - Intro to Financial Math
Page: 4 of 5
D. FALSE
The profit of the stock investor is always by 100e1.1 − 100 > 0 below the profit of the
forward-contract investor.
E. TRUE
Without any dividend payments, the two positions are equivalent as we saw in our
analysis of the offered choice C. The dividend payment is given to the stock investor
and not to the forward investor, so the stock investor is certainly better off.
3.2.3. Recommended problems. McDonald: #2.7, #2.8, #2.9.
3.3. Hedging with forwards.
3.3.1. Selling forward. The following graph shows the payoff curves for the producer’s position if (s)he decideds to hedge by selling forward, i.e., entering a short forward contract.
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The blue line corresponds to the producer’s inherent long position in the good (s)he
produces; this is the unhedged position. The red line is the payoff curve for the short
forward contract (with the forward price of 500 chosen arbitrarily for the purpose of being
able to produce a graph). The forward contract is the hedge. The sum of the two payoffs is
the payoff of the producer’s entire hedged portfolio. It is the gold line in the figure above.
As we can see, the producer completely removed the randomness from his/her portfolio.
The payoff of the hedged portfolio is “flat” at the forward price. The profit of the hedged
portfolio is, hence, constant at the difference between the forward price and the total (fixed
and variable) costs.
3.3.2. Buying forward. As we have already noted above, the buyer/user of goods/services
has an inherent short position in that particular commodity.
Example 3.3. Pancakes, Inc. produces a variety of pancake products. It longed forward
contract on 100 lbs of blueberries at $2.00 per pound. Revenue is $1, 000 for the pancakes
produced with the above blueberries. Other costs total $700. Find the profit.
Instructor: Milica Čudina
Lecture: 3
Course: M339D/M389D - Intro to Financial Math
Page: 5 of 5
Solution:
1000 − 200 − 700 = 100
The following graph shows the profit curves for the buyer’s position if (s)he decideds to
hedge by endering a long forward contract.
600
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200
200
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600
800
1000
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The blue line is the profit curve based on the fixed revenue (reduced by additional total
costs) of 700; this is the unhedged position. The red line is the payoff/profit curve for the
long forward contract (with the forward price of 500). The forward contract is the hedge.
The sum of the two profits is the profit of the buyer’s entire hedged portfolio. It is the gold
line in the figure above. Again, there is nothing random about the buyer’s hedged portoflio’s
profit.
Remark 3.4. An asset-investor with a long/short position in the underlying also hedges using
a short/long forward contract. The entire payoff/profit structure is completely analogous to
the situations discussed above.
Instructor: Milica Čudina